SUPERCONVERGENCE OF THE VELOCITY IN ... - CNLS, LANL

c 2005 Society for Industrial and Applied Mathematics

SIAM J. NUMER. ANAL. Vol. 43, No. 4, pp. 1728–1749

SUPERCONVERGENCE OF THE VELOCITY IN MIMETIC FINITE DIFFERENCE METHODS ON QUADRILATERALS∗ M. BERNDT† , K. LIPNIKOV† , M. SHASHKOV† , M. F. WHEELER‡ , AND I. YOTOV§ Abstract. Superconvergence of the velocity is established for mimetic finite difference approximations of second-order elliptic problems over h2 -uniform quadrilateral meshes. The superconvergence result holds for a full tensor coefficient. The analysis exploits the relation between mimetic finite differences and mixed finite element methods via a special quadrature rule for computing the scalar product in the velocity space. The theoretical results are confirmed by numerical experiments. Key words. mixed finite element, mimetic finite difference, tensor coefficient, superconvergence AMS subject classifications. 65N06, 65N12, 65N15, 65N22, 65N30 DOI. 10.1137/040606831

1. Introduction. We consider the numerical approximation of a linear secondorder elliptic problem. In porous medium applications, this equation models single phase Darcy flow and is usually written as a first-order system for the fluid pressure p and velocity u: (1.1)

u = −K grad p div u = f u·n = g

in Ω, in Ω, on ∂Ω,

where Ω ⊂ 2 , n is the outward unit normal to ∂Ω, and K ∈ 2×2 is a symmetric uniformly positive definite full tensor representing the rock permeability divided by the fluid viscosity. We assume that system (1.1) satisfies the compatibility condition f dx + g ds = 0. Ω

∂Ω

In this paper, we analyze the convergence of a mimetic finite difference (MFD) method on quadrilateral meshes. The method uses discrete operators that preserve certain critical properties of the original continuum differential operators. Conservation laws, solution symmetries, and the fundamental identities and theorems of vector ∗ Received by the editors April 15, 2004; accepted for publication (in revised form) February 7, 2005; published electronically November 22, 2005. This work was performed by an employee of the U.S. Government or under U.S. Government contract. The U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes. Copyright is owned by SIAM to the extent not limited by these rights. http://www.siam.org/journals/sinum/43-4/60683.html † Mathematical Modeling and Analysis Group, Theoretical Division, Mail Stop B284, Los Alamos National Laboratory, Los Alamos, NM 87545 ([email protected], [email protected], shashkov@ lanl. gov). The research of these authors was supported by the U.S. Department of Energy under contract W-7405-ENG-36, LA-UR-03-7904. ‡ ICES, Department of Aerospace Engineering and Engineering Mechanics, Department of Petroleum and Geosystems Engineering, and Department of Mathematics, The University of Texas at Austin, Austin, TX 78712 ([email protected]). The work of this author was partially supported by NSF grant EIA-0121523 and by NPACI grant UCSD 10181410. § Department of Mathematics, 301 Thackeray Hall, University of Pittsburgh, Pittsburgh, PA 15260 ([email protected]). The work of this author was partially supported by NSF grants DMS 0107389, DMS 0112239, and DMS 0411694 and by DOE grant DE-FG02-04ER25618.

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SUPERCONVERGENCE OF THE VELOCITY

1729

and tensor calculus are examples of such properties. This “mimetic” technique has been applied successfully to several applications including diffusion [22, 15, 18], magnetic diffusion and electromagnetics [14], continuum mechanics [17], and gas dynamics [8]. For problem (1.1), the mimetic technique uses discrete flux G and divergence DIV operators for the continuum operators −Kgrad and div, respectively, which are adjoint to each other, i.e., G = DIV ∗ . It is straightforward to extend the MFD method to locally refined meshes with hanging nodes [16], unstructured three-dimensional meshes composed of hexahedra, tetrahedra, and any cell type having three faces intersecting at each vertex. A connection between the MFD method and the mixed finite element (MFE) method with Raviart–Thomas finite elements has been established in [4]. In particular, it was shown that the scalar product in the velocity space proposed in [15] for MFD methods can be viewed as a quadrature rule in the context of MFE methods. Another closely related method is the control-volume MFE method [7, 9]. MFE discretizations on quadrilateral grids have been studied in [25, 26, 2, 13]. These methods are based on the Piola transformation [25, 6], which preserves continuity of the normal component of the velocity u across mesh edges. Unfortunately, this results in the necessity to integrate rational functions over quadrilaterals. The task becomes even more complicated when the diffusion tensor is full and nonconstant. The results in [4] provide an efficient numerical quadrature rule with a minimal number of points. Moreover, the connection between the two methods allows for extensions of MFE methods to general polygons and polyhedra. The aforementioned connection provides a suitable functional frame for rigorous analysis of convergence of mimetic discretizations. In [4], first-order convergence for the fluid pressure and velocity was shown. In this paper, we establish velocity superconvergence for MFD discretizations of (1.1) on h2 -uniform quadrilateral meshes (as defined in (2.2)–(2.3)). Precise calculation of the fluid velocity is important for porous media and other applications. The points or lines where the numerical solution is superclose to the exact solution may be used to improve the accuracy of the overall simulation. Various superconvergence results for MFE methods have been established for rectangular meshes [21, 19, 27, 10, 11, 12, 3, 1] and general quadrilateral meshes [2, 13]. In [13], velocity superconvergence is established for the MFE discretization of (1.1) on h2 -uniform quadrilateral grids. In this paper, we exploit the relation between MFD methods and MFE methods with the quadrature rule (3.10) to establish superconvergence for velocities in MFD discretizations. In particular, we show that the computed normal velocities are superclose to the true normal velocities at the midpoints of the edges. In [18], an alternative quadrature is introduced, which preserves symmetry of the exact solution on polar grids. This symmetry preservation is important for problems of radiation transport in the asymptotic diffusion limit. The analysis of superconvergence for symmetry-preserving quadratures is left for future investigation. The paper outline is as follows. In section 2, we describe the MFE method for (1.1). In section 3, the MFD method is presented and related to the MFE method with a quadrature rule. The main superconvergence results are presented in section 4. Superconvergence of the normal velocities at the midpoints of the edges is established in section 5. In section 6, numerical experiments are given that confirm the theoretical results.

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BERNDT, LIPNIKOV, SHASHKOV, WHEELER, AND YOTOV

r3

r4 r3 =

r4

r2 r1 r2 = r1 Fig. 2.1. h2 -uniform quadrilateral grid.

2. The MFE method. To simplify the exposition, we assume without loss of generality that g = 0, i.e., homogeneous Neumann boundary conditions are imposed on ∂Ω. Throughout this paper, we shall use the notation · k,D , · div ,D, and · D for the norms on the Hilbert spaces H k (D), H(div; D), and L2 (D), respectively, where D ⊂ Ω. In addition, | · |k,D will denote the seminorm on H k (D). To simplify notation, we shall omit the subscript D when D = Ω. Finally, we denote by (·, ·) the L2 -inner product on Ω of either scalar or vector functions. Let V = {v ∈ H(div; Ω) : v · n = 0 on ∂Ω} and W = w ∈ L2 (Ω) : w dx = 0 . Ω

The variational formulation of (1.1) is as follows: find a pair (u, p) ∈ V × W such that

(2.1)

(K−1 u, v) − (p, div v) = 0, (div u, w) = (f, w)

∀ (v, w) ∈ V × W.

For the discretization of (2.1), denote by Th a shape-regular partition (see [5, ¯ into convex quadrilateral elements of diameter not greater Remark 2.2, p. 113]) of Ω than h. For two examples of shape-regular grids, see Figure 6.1. We assume that the grid is h2 -uniform. Following [13], the quadrilateral partition Th is called h2 -uniform if each element is an h2 -parallelogram, i.e., (2.2)

(r2 − r1 ) − (r3 − r4 ) ≤ Ch2 ,

and any two adjacent quadrilaterals form an h2 -parallelogram, i.e., (2.3)

(r2 − r1 ) − (r2 − r1 ) ≤ Ch2 ,

where r1 , r2 , r3 , and r4 are the vertices of the adjacent element (see Figure 2.1). For any convex quadrilateral e, there exists a bijection mapping Fe : eˆ → e, where eˆ is the reference unit square with vertices ˆ r1 = (0, 0)T , ˆ r2 = (1, 0)T , ˆ r3 = (1, 1)T , T T and ˆ r4 = (0, 1) . Denote by ri = (xi , yi ) , i = 1, 2, 3, 4, the four corresponding vertices of element e as shown in Figure 2.2. Then, Fe is the bilinear mapping given by (2.4)

Fe (ˆ r) = r1 (1 − x ˆ)(1 − yˆ) + r2 x ˆ(1 − yˆ) + r3 x ˆyˆ + r4 (1 − x ˆ)ˆ y.

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ˆ3 n

n3

ˆr4

n2 r4

ˆ3

ˆ4 n

ˆ4

ˆr1

r3

ˆr3

eˆ

2

e

ˆ2 n

ˆ2

ˆ1

3

Fe

4

n4

r2 1

ˆr2

n1

r1

ˆ1 n Fig. 2.2. Bilinear mapping and orientation of normal vectors.

Note that the Jacobi matrix DFe and its Jacobian Je are linear functions of x ˆ and yˆ. Indeed, straightforward computations yield (2.5)

DFe = [(1 − yˆ) r21 + yˆ r34 , (1 − x ˆ) r41 + x ˆ r32 ]

and (2.6)

Je = 2|T124 | + 2(|T123 | − |T124 |)ˆ x + 2(|T134 | − |T124 |)ˆ y,

where rij = ri − rj and |Tijk | is the area of the triangle with vertices ri , rj , and rk . Since e is convex, the Jacobian Je is always positive, i.e., Je > 0. ˆ i be Let i and î , i = 1, 2, 3, 4, be the edges of e and eˆ, respectively. Let ni and n the unit outward normal vectors to i and î , respectively (see Figure 2.2). Similarly, let τ i and τˆ i be the unit tangential vectors to i and î , respectively. It is easy to see from (2.5) that for any edge i , (2.7)

ni =

1 î Je DF−T e n |i |

and

τi =

1 DFe τˆ i . |i |

The reader is referred to [6] for suitable choices for the pair of finite element spaces Vh ⊂ V and W h ⊂ W . In this paper, we consider the lowest-order Raviart–Thomas finite element spaces RT0 [25, 20] defined on the reference element eˆ as ˆ e) = P1,0 (ˆ e) × P0,1 (ˆ e), V(ˆ

ˆ (ˆ W e) = P0 (ˆ e),

where P1,0 (or P0,1 ) denotes the space of polynomials linear in the x ˆ (or yˆ) variable and constant in the other variable, and P0 denotes the space of constant functions. The velocity space on any convex quadrilateral e is defined through the Piola transformation [6] 1 DFe : L2 (ˆ e) × L2 (ˆ e) → L2 (e) × L2 (e) Je

∀e ∈ Th .

The RT0 spaces on Th are given by (2.8)

ˆ e) ∀e ∈ Th }, Vh = {v ∈ V : v|e = Je−1 DFe v ˆ ◦ F−1 ˆ ∈ V(ˆ e , v ˆ (ˆ ˆ ◦ F−1 , w ˆ∈W e) ∀e ∈ Th }. W h = {w ∈ W : w|e = w e

Two properties of Piola’s transformation will be important in our analysis. For any ˆ e) and the related v = Je−1 DFe v ˆ ◦ F−1 v ˆ ∈ V(ˆ e , (2.9)

v Je div v = div ˆ

and

ˆ i. |i | v · ni = v ˆ·n

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Note that since Vh ⊂ H(div; Ω), any vector in Vh has continuous normal components on the edges. A function in W h is uniquely determined by its values at the cell-centers and a vector in Vh is uniquely determined by its normal components on the edges. Therefore, dimW h = Np and dimVh = Ne , where Np is the number of elements and Ne is the number of interior edges. Let {ψih }, i = 1, Np , be a basis for W h such that 1, i = j, h ψi (cj ) = δij ≡ 0, i = j, where cj is the center of element ej , j = 1, Np . Similarly, let φhi , i = 1, Ne , be a basis for Vh such that φhi · nj = δij , where nj is a fixed unit normal vector on edge j , j = 1, Ne . In order to simplify notation, we use the same way for global and local indexing of mesh edges and corresponding normal vectors. Given the finite element spaces Vh and W h , we define the discrete problem: find h (u , ph ) ∈ Vh × W h such that (2.10)

(K−1 uh , vh )h − (ph , div vh ) = 0, (div uh , wh ) = (f, wh )

∀ (vh , wh ) ∈ Vh × W h ,

where (·, ·)h is a continuous bilinear form corresponding to the application of a numerical quadrature rule for computing (·, ·). A detailed discussion of this quadrature rule is given in section 3. 3. MFD discretizations. In this section, we derive an MFD discretization of (1.1) and show its connection with the MFE method (2.10). The first step in the mimetic technique is to specify discrete degrees of freedom for pressure and velocity. The discrete pressure unknowns are defined at the centers of the quadrilaterals, one unknown per mesh cell. The discrete velocities are defined at the midpoints of mesh edges as normal components. In other words, an edge-based unknown is a scalar and represents the orthogonal projection of a velocity vector onto the unit vector ni normal to the mesh edge i . The second step in the mimetic technique is to equip the spaces of discrete pressures and velocities with scalar products. We denote the vector space of cell-centered pressures by Qd . The dimension of Qd equals the number of mesh cells Np . The scalar product on the vector space Qd is given by (3.1)

d

d

[p , q ]Qd =

Np

|ei | pdi qid

∀pd , q d ∈ Qd ,

i=1

where |ei | denotes the area of cell ei and pdi , qid are cell-centered pressure components. It is easy to see that the vector space Qd is isometric to the MFE space W h in (2.8). Indeed, for any ph ∈ W h , there exists a unique pd = (pd1 , pd2 , . . . , pdNp )T ∈ Qd Np d h such that ph = i=1 pi ψi and (ph , q h ) = [pd , q d ]Qd . Note that the discrete MFD pressure variable, pdi , corresponds to the value of the MFE pressure function at the cell-center, ph (ci ).

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We denote the vector space of edge-based velocities by X d . The dimension of X d equals the number of interior mesh edges Ne . The scalar product on X d is given by [ud , vd ]X d = (3.2) [ud , vd ]X d ,e , e∈Th

where [ud , vd ]X d ,e is a scalar product over cell e involving only the normal velocity components on cell edges. Recall that a velocity vector can be recovered from two orthogonal projections on any two noncollinear vectors. Since the mesh cell is convex, any pair of normal vectors to edges with a common point satisfies the above requirement. The orthogonal projections are exactly the degrees of freedom associated with cell edges. As shown in Figure 3.1, four recovered velocity vectors can be associated with the four vertices of the quadrilateral. For example, velocity v1 is recovered from its projections onto the normal vectors n1 and n2 . For a general quadrilateral e, we denote by vd (rj ) the velocity recovered at jth vertex rj , j = 1, 2, 3, 4. Then, the cell-based scalar product is given by 1 |Tj | K−1 (rj )ud (rj ) · vd (rj ), 2 j=1 4

[ud , vd ]X d ,e =

(3.3)

where |Tj | is the area of the triangle with vertices rj−1 , rj , and rj+1 (see Figures 2.2 and 3.1). For example, triangles T1 and T4 are the shaded triangles in Figure 3.1. Note that (3.3) is indeed an inner product, since K is a symmetric and positive definite tensor and [vd , vd ]X d ≥ C|||vd |||2 ,

(3.4)

where ||| · ||| is the Euclidean vector norm. n4 v4 T4 T1

n1 n1

n2 v1 Fig. 3.1. Recovered vectors v1 , v4 and triangles T1 , T4 .

The vector space X d is isomorphic to the MFE space Vh in (2.8), since both spaces have the same definitions of degrees of freedom. In particular, for any vh ∈ Vh , Ne d d d d T d h there exists a unique v = (v1 , v2 , . . . , vN e ) ∈ X such that v = i=1 vid φhi . Note d that the discrete MFD velocity variable, vi , corresponds to the MFE normal velocity component, vh · ni , on edge i . The third step in the mimetic technique is to derive a discrete approximation to the divergence operator, DIV : X d → Qd , which we shall refer to as the prime

1734


operator. For a cell e, the Gauss divergence theorem gives (3.5)

DIV ud |e =

1 d u1 |1 | + ud2 |2 | + ud3 |3 | + ud4 |4 | , |e|

where ud1 , . . . , ud4 are the normal velocity components on element e and the normal vectors are oriented as shown in Figure 2.2. The fourth step in the mimetic technique is to derive a discrete flux operator G (for the continuous operator −Kgrad ) adjoint to the discrete divergence operator DIV with respect to scalar products (3.1) and (3.2), i.e., [DIV ud , pd ]Qd ≡ [ud , Gpd ]X d

∀ud ∈ X d

∀pd ∈ Qd .

To derive the explicit formula for G, we consider an auxiliary scalar product · , · and relate it to scalar products (3.1) and (3.2). Denote by · , · the standard vector dot product. Then [pd , q d ]Qd = Dpd , q d

[ud , vd ]X d = Mud , vd ,

and

where D is a diagonal matrix, D = diag{|e1 |, . . . , |eNp |}, and M is a sparse symmetric mass matrix with a 5-point stencil. Restricted to a cell, this stencil connects edge-based unknowns if and only if the corresponding edges have a common point. Combining the last two formulae, we get [ud , DIV ∗ pd ]X d = ud , M DIV ∗ pd = [DIV ud , pd ]Qd = ud , DIV t D pd ∀ud ∈ X d ∀pd ∈ Qd , where DIV t is the adjoint of DIV with respect to the auxiliary scalar product. Therefore, G = M−1 DIV t D.

(3.6)

The MFD method approximating first-order system (1.1) may be summarized as follows: (3.7)

ud = G pd ,

DIV ud = f d ,

d t ) , and entry fid is the integral average of f over cell ei . where f d = (f1d , . . . , fN p The basic tool for the error analysis of the discrete solution (ud , pd ) ∈ X d × Qd is based on the following transformation. Multiplying the first equation in (3.7) by Mvd and the second one by Dq d , we get

(3.8)

[ud , vd ]X d − [pd , DIV vd ]Qd = 0, [q d , DIV ud ]Qd = [f d , q d ]Qd

∀(vd , q d ) ∈ X d × Qd .

Using the isomorphism between the finite element space Vh ×W h and the vector space X d × Qd , we define finite element functions ph , q h , f h , uh , and vh corresponding to vectors pd , q d , f d , ud , and vd , respectively. Then [pd , DIV vd ]Qd = (ph , div vh )

and

[q d , DIV ud ]Qd = (q h , div uh ).

The definition of f d implies that [f d , q d ]Qd = (f h , q h ) = (f, q h ).


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Finally, by introducing the quadrature rule (K−1 uh , vh )h ≡ [ud , vd ]X d ,

(3.9)

we reduce problem (3.7) to the finite element problem (2.10). The scalar product in the space of velocities given by (3.3) is obviously not unique. In the context of MFE methods, it is a quadrature rule for numerical integration of (K−1 uh , vh ): 1 |Tj | K−1 (rj )uh (rj ) · vh (rj ), 2 j=1 4

(3.10)

(K−1 uh , vh )h,e =

where uh (rj ) is the recovered velocity at vertex rj . In the context of MFE methods, we shall refer to (3.10) as the MFD quadrature rule. The global scalar product is obtained by summing over quadrilaterals, i.e., (3.11) (K−1 uh , vh )h,e . (K−1 uh , vh )h = e∈Th

Note that (3.4) implies that there exists a constant C0 > 0 such that (3.12)

(K−1 vh , vh )h ≥ C0 vh 2

∀vh ∈ Vh .

It was shown in [4] that the element quadrature rule (3.10) is exact for any constant vector uh , constant tensor K, and vh ∈ Vh . 4. Superconvergence estimates for the velocity. We begin by recalling the mixed projection operator Π : H 1 (Ω) × H 1 (Ω) → Vh satisfying (4.1)

(div (Π v − v), w) = 0

∀w ∈ W h .

The operator Π is defined locally on each element e by

= Πˆ ˆ v, Πv ˆ : H 1 (ˆ ˆ e) is the reference element projection operator satiswhere Π e) × H 1 (ˆ e) → V(ˆ fying ˆv − v ˆ i = 0, i = 1, 2, 3, 4. (4.2) (Πˆ ˆ) · n î

The approximation properties of Π have been established in [25, 26]: (4.4)

Πvdiv ≤ Cv1 , Πv − v ≤ Chv1 ,

(4.5)

div (Πv − v) ≤ Chv2 .

(4.3)

ˆ which will be used The following lemma gives several approximation properties of Π in the analysis. ˆ defined in (4.2) satisfies, for any v Lemma 4.1. The operator Π ˆ = (ˆ v1 , vˆ2 ) in 1 1 H (ˆ e) × H (ˆ e), the following: ∂ ˆ ∂ ˆ (4.6) x dˆ y = 0, x dˆ y = 0, (Πˆ v−v ˆ)1 dˆ (Πˆ v−v ˆ)2 dˆ ∂ x ˆ ∂ ˆ eˆ eˆ y ∂ ˆ v)1 ≤ C ∂ vˆ1 , ∂ (Πˆ ˆ v)2 ≤ C ∂ vˆ2 , (Πˆ (4.7) ∂x ˆ ∂x ˆ ∂ yˆ ∂ yˆ eˆ eˆ eˆ eˆ (4.8)

ˆ v1,ê ≤ Cˆ v1,ê . Πˆ

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Proof. The identities in (4.6) follow easily from definition (4.2). In particular, writing (4.2) for the two vertical edges gives 1 1 ˆv − v ˆv − v (Πˆ ˆ)1 (0, yˆ) dˆ y = 0, (Πˆ ˆ)1 (1, yˆ) dˆ y = 0. 0

0

Subtracting the above equations and applying the fundamental theorem of calculus implies the first identity in (4.6). The proof of the second identity is similar. Note ˆ v)1 and ∂ (Πˆ ˆ v)2 are the L2 -orthogonal projections of that (4.6) means that ∂∂xˆ (Πˆ ∂ yˆ ∂ ∂ ˆ1 and ∂ yˆ vˆ2 , respectively, onto the space of constants, which implies (4.7). Finally, ∂x ˆv it is easy to see that (4.2) implies ˆ veˆ ≤ Cˆ v1,ê , Πˆ which, combined with (4.7), gives (4.8). We also make use of the L2 -projection operator Ph : W → W h such that for p ∈ W, (4.9)

(Ph p − p, w) = 0

∀w ∈ W h .

Denote the quadrature error by (4.10)

σ(q, v) ≡ (q, v) − (q, v)h .

The variational formulation (2.1) and the discrete problem (2.10) give rise to the error equations (K−1 (Πu − uh ), vh )h = (Ph p − ph , div vh ) (4.11)

+ (K−1 (Πu − u), vh ) − σ(K−1 Πu, vh ), (div (Πu − uh ), wh ) = 0,

where we used (4.9) and (4.1) in the first and second equations, respectively. We note that, using (2.9), the second equation in (4.11) gives (Πˆ û − u ˆ h ), w ˆ h )eˆ 0 = (div (Πu − uh ), wh )e = (div

∀wh ∈ Wh .

V (Πˆ (Πˆ ˆh = W ˆ h , taking w û − u û − u ˆ h ) implies that div ˆ h ) = 0 and Since div ˆ h = div therefore, by (2.9), (4.12)

div (Πu − uh ) = 0.

Taking vh = Πu − uh ∈ Vh and wh = Ph p − ph in (4.11) gives (4.13) (K−1 (Πu − uh ), Πu − uh )h = (K−1 (Πu − u), Πu − uh ) − σ(K−1 Πu, Πu − uh ). The estimate for the first term on the right-hand side of (4.13) follows from Theorem 5.1 in [13] and (4.12):

(4.14)

(K−1 (Πu − u), Πu − uh ) ≤ C h2 u2 Πu − uh + u1 div (Πu − uh ) = C h2 u2 Πu − uh .


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The second term on the right-hand side of (4.13) can be bounded using Lemma 4.3: (4.15)

|σ(K−1 Πu, Πu − uh )| ≤ C h2 u2 Πu − uh .

Combining (4.14), (4.15), and (3.12), we obtain the following superconvergence result. Theorem 4.2. Let K−1 ∈ W 2,∞ (Ω). For the velocity uh of the MFE method (2.1), on h2 -uniform quadrilateral grids, there exists a positive constant C independent of h such that Πu − uh ≤ C h2 u2 .

(4.16)

We now proceed to prove estimate (4.15). Lemma 4.3. Let v ∈ Vh , and let K−1 ∈ W 2,∞ (Ω). There exists a positive constant C independent of h such that (4.17)

|σ(K−1 Πu, v)| ≤ C h2 (u2 v + u1 div v).

Proof. For an element e ∈ T h , we define the error −1 σe (K Πu, v) = K−1 Πu · v dx − (K−1 Πu, v)h,e . (4.18) e

With (3.10), the second term on the right-hand side of (4.18) can be written as (4.19) 1 |Tj |K−1 (rj )Πu(rj ) · v(rj ) 2 j=1 4

(K−1 Πu, v)h,e =

1 1 ˆ u (ˆ DFe Πˆ DFe v ˆ (ˆ rj ) rj ) · Je Je

=

1 ˆ −1 (ˆ |Tj |K rj ) 2 j=1

=

1 1 |Tj | ˆ −1 (ˆ ˆ u(ˆ DFTe (ˆ rj )K rj )DFe (ˆ rj ) Πˆ rj ) · v ˆ(ˆ rj ) 2 j=1 Je (ˆ rj ) Je (ˆ rj )

=

1 ˆ u(ˆ Be (ˆ rj ) Πˆ rj ) · v ˆ(ˆ rj ) 4 j=1

4

4

4

ˆ u, v ≡ (Be Πˆ ˆ)T , where the subscript T denotes the trapezoidal rule on element eˆ and we define Be = |T | T ˆ −1 1 DFe . Here we used (2.6) to conclude that Je (ˆrj j ) = 12 . Considering the Je DFe K first term on the right-hand side of (4.18), we obtain ˆ u · 1 DFe v ˆ −1 1 DFe Πˆ ˆ K−1 Πu · v dx = K ˆJe d x Je Je e eˆ 1 û · v ˆ −1 DFe Πˆ ˆ = DFTe K ˆ dx (4.20) J eˆ e û · v ˆ. ˆ dx = Be Πˆ eˆ

Substituting (4.19) and (4.20) into (4.18), we obtain −1 û · v ˆ u, v ˆ u, v ˆ − Be Πˆ σe (K Πu, v) = Be Πˆ (4.21) ˆ dx ˆ T ≡ σeˆ Be Πˆ ˆ . eˆ

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Hereafter we shall omit the subscripts e and eˆ. Let E(f ) ≡ f (ˆ x, yˆ)dˆ x dˆ y − (f )T eˆ

be the error of the trapezoidal rule for integrating a function f (ˆ x, yˆ) on eˆ. Then, ˆ u)2 vˆ2 . ˆ u, v ˆ u)1 vˆ1 + E (BΠˆ (4.22) σ(BΠˆ ˆ) = E (BΠˆ We next bound the first term on the right-hand side in (4.22). The argument for the bound on the second term is similar. Using the trapezoidal rule error representation from Lemma A.1 based on the Peano kernel theorem (see [23, Theorem 5.2-3, p. 142]), we write

1

ˆ u)1 vˆ1 ) = E((BΠˆ

1

φ(ˆ x) 0

0 1 1

∂2 ˆ (BΠˆ u)1 vˆ1 (ˆ x, 0) dˆ x dˆ y ∂x ˆ2

∂2 ˆ u)1 (0, yˆ)ˆ (BΠˆ v1 (0, yˆ) dˆ x dˆ y ∂ yˆ2 0 0 1 1 ∂2 ˆ (BΠˆ u)1 vˆ1 (ˆ + x, yˆ) dˆ x dˆ y ψ(ˆ x, yˆ) ∂x ˆ∂ yˆ 0 0 ≡ (I) + (II) + (III), +

(4.23)

φ(ˆ y)

where φ(t) = t(t − 1)/2 and ψ(s, t) = (1 − s)(1 − t) − 1/4. Denote by B11 , B12 , B21 , and B22 the components of the tensor B. Since vˆ1 (0, yˆ) is constant in yˆ, the second term in (4.23) is

1

1

(II) =

φ(ˆ y) 0

0 1 1

+

(4.24)

∂2 ˆ u)1 (0, yˆ)ˆ B11 (0, yˆ)(Πˆ v1 (0, yˆ) dˆ x dˆ y ∂ yˆ2

φ(ˆ y) 0

0 1

∂2 ˆ u)2 (0, yˆ)ˆ B12 (0, yˆ)(Πˆ v1 (0, yˆ) dˆ x dˆ y ∂ yˆ2

1

∂ ∂ ˆ B12 (0, yˆ) (Πˆ u)2 (0, yˆ)ˆ v1 (0, yˆ) dˆ x dˆ y ∂ y ˆ ∂ yˆ 0 0 ≡ (II)1 + (II)2 + (II)3 . +2

φ(ˆ y)

Using (4.8), for the first two terms on the right-hand side, we have u1,ê ˆ v1 eˆ. |(II)1 | + |(II)2 | ≤ C|B|2,∞,ê ˆ Since

∂ ˆ u)2 ∂ yˆ (Πˆ

is a constant, we rewrite the last term in (4.24) as

1

1

∂ ∂ ˆ u)2 (ˆ φ(ˆ y ) B12 (0, yˆ) (Πˆ x, yˆ)ˆ v1 (0, yˆ) dˆ x dˆ y ∂ y ˆ ∂ yˆ 0 0 ∂ ˆ u)2 ˆ ≤ C|B|1,∞,ê ( Πˆ u|1,ê ˆ v1 eˆ, ∂ yˆ v1 eˆ ≤ C|B|1,∞,ê |ˆ eˆ

(II)3 = 2

using (4.7). A combination of the last two bounds implies that v1 eˆ. (4.25) u1,ê + |B|1,∞,ê |ˆ u|1,ê ˆ |(II)| ≤ C |B|2,∞,ê ˆ


1739

x, yˆ) is constant in yˆ, we have For the last term in (4.23), since vˆ1 (ˆ 1 1 ∂2 ˆ u)1 (ˆ (4.26) (III) = ψ(ˆ x, yˆ) x, yˆ)ˆ v1 (ˆ x, yˆ) dˆ x dˆ y (BΠˆ ∂x ˆ∂ yˆ 0 0 1 1 ∂ ∂ ˆ u)1 (ˆ + ψ(ˆ x, yˆ) (BΠˆ x, yˆ) vˆ1 (ˆ x, yˆ) dˆ x dˆ y ≡ (III)1 + (III)2 . ∂ yˆ ∂x ˆ 0 0 Using (4.7), |(III)1 | ≤ C(|B|2,∞,ê ˆ u1,ê + |B|1,∞,ê |ˆ u|1,ê )ˆ v1 eˆ.

(4.27)

11 x, yˆ) is a constant and 0 0 ψ(ˆ x, yˆ) dˆ x dˆ y = 0. To bound (III)2 we note that ∂∂xˆ vˆ1 (ˆ Therefore, by the Bramble–Hilbert lemma [5], and using (4.7), ∂ ˆ u)1 ˆ v1 eˆ ≤ C(|B|2,∞,ê ˆ (4.28) |(III)2 | ≤ C (BΠˆ u1,ê + |B|1,∞,ê |ˆ u|1,ê )ˆ v1 eˆ. ∂ yˆ 1,ˆ e The first term in the error representation (4.23) is 1 1 ∂2 ˆ u)1 (ˆ (I) = (4.29) φ(ˆ x) 2 (BΠˆ x, 0) vˆ1 (ˆ x, 0) dˆ x dˆ y ∂x ˆ 0 0 1 1 ∂ ∂ ˆ u)1 (ˆ +2 φ(ˆ x) (BΠˆ x, 0) vˆ1 (ˆ x, 0) dˆ x dˆ y = (I)1 + (I)2 . ∂ x ˆ ∂ x ˆ 0 0 The first term on the right-hand side can be bounded in a way similar to (II): |(I)1 | ≤ C(|B|2,∞,ê ˆ u1,ê + |B|1,∞,ê |ˆ u|1,ê )ˆ v1 eˆ.

(4.30)

We rewrite the second term on the right-hand side in (4.29) as (4.31) 1 (I)2 = 2

1

1

φ(ˆ x) 0

0 1 1

+

φ(ˆ x) 0

0

∂ ∂ ∂ ˆ u)1 (ˆ ˆ u)1 (ˆ (BΠˆ (BΠˆ vˆ1 (ˆ x, 0) − x, yˆ) x, 0) dˆ x dˆ y ∂x ˆ ∂x ˆ ∂x ˆ ∂ ∂ ˆ u)1 (ˆ x, yˆ) vˆ1 (ˆ x, 0) dˆ x dˆ y ≡ (I)2,1 + (I)2,2 . (BΠˆ ∂x ˆ ∂x ˆ

To estimate the first term in (4.31), we write ∂ ∂ ˆ u)1 (ˆ ˆ u)1 (ˆ x, yˆ) − x, 0) = (BΠˆ (BΠˆ ∂x ˆ ∂x ˆ

0

yˆ

∂2 ˆ u)1 (ˆ x, tˆ) dtˆ. (BΠˆ ∂x ˆ∂ yˆ

This allows us to bound the first term in (4.31) in a way similar to bounds (4.25) and (4.30): (4.32)

|(I)2,1 | ≤ C(|B|2,∞,ê ˆ u1,ê + |B|1,∞,ê |ˆ u|1,ê )ˆ v1 eˆ.

The second term on the right-hand side in (4.31) can be rewritten as 1 1 ∂ ∂ û − u ˆ ))1 (ˆ (4.33) (I)2,2 = φ(ˆ x) (B(Πˆ x, yˆ) vˆ1 (ˆ x, yˆ) dˆ x dˆ y ∂x ˆ ∂x ˆ 0 0 1 1 ∂ ∂ + φ(ˆ x) (Bˆ x, yˆ) vˆ1 (ˆ x, yˆ) dˆ x dˆ y ≡ (I)2,2,1 + (I)2,2,2 , u)1 (ˆ ∂x ˆ ∂x ˆ 0 0

1740


where we used that ∂∂xˆ vˆ1 (ˆ x, 0) = ∂∂xˆ vˆ1 (ˆ x, yˆ) on e, since vˆ1 is a constant in yˆ. To estimate the second term in (4.33), we use the identity ∂ ∂ v vˆ1 = − vˆ2 + div ˆ. ∂x ˆ ∂ yˆ

(4.34) We rewrite (I)2,2,2 as (4.35)

1 1 ∂ ∂ ∂ v (Bˆ u)1 x dˆ y+ φ(ˆ x) ˆ dˆ x dˆ y vˆ2 dˆ (Bˆ u)1 div ∂ x ˆ ∂ y ˆ ∂ x ˆ 0 0 0 0 1 1 ∂ ∂2 (Bˆ u)1 vˆ2 dˆ = − φ(ˆ x) x+ φ(ˆ x) x dˆ y (Bˆ u)1 vˆ2 dˆ ∂x ˆ ∂x ˆ∂ yˆ 0 0 ˆ1 ˆ3 1 1 ∂ v + φ(ˆ x) ˆ dˆ x dˆ y. (Bˆ u)1 div ∂x ˆ 0 0

(I)2,2,2 = −

1

1

φ(ˆ x)

Clearly, the last two terms can be bounded by (4.36)

v v2 eˆ + |(Bˆ u)1 |1,ê div êˆ). C(|(Bˆ u)1 |2,ê ˆ

We postpone the estimate of the edge integrals in (4.35) for later. To bound the first term on the right-hand side in (4.33), we have 1 1 ∂ ∂ û − u (I)2,2,1 = φ(ˆ x) B11 (ˆ x, yˆ)(Πˆ ˆ )1 (ˆ x, yˆ) vˆ1 (ˆ x, yˆ) dˆ x dˆ y ∂x ˆ ∂x ˆ 0 0 1 1 ∂ ˆ ∂ + u−u ˆ )1 (ˆ φ(ˆ x)B11 (ˆ x, yˆ) (Πˆ x, yˆ) vˆ1 (ˆ x, yˆ) dˆ x dˆ y ∂x ˆ ∂x ˆ 0 0 1 1 (4.37) ∂ ∂ û − u + φ(ˆ x) B12 (ˆ x, yˆ)(Πˆ ˆ )2 (ˆ x, yˆ) vˆ1 (ˆ x, yˆ) dˆ x dˆ y ∂ x ˆ ∂ x ˆ 0 0 1 1 ∂ ˆ ∂ + u−u ˆ )2 (ˆ φ(ˆ x)B12 (ˆ x, yˆ) (Πˆ x, yˆ) vˆ1 (ˆ x, yˆ) dˆ x dˆ y ∂ x ˆ ∂ x ˆ 0 0 ≡ (I)2,2,1,1 + (I)2,2,1,2 + (I)2,2,1,3 + (I)2,2,1,4 . ˆ u is exact for constants, using the Bramble–Hilbert lemma and the inverse Since Πˆ inequality, we can bound the first and the third terms in (4.37) as (4.38)

|(I)2,2,1,1 | + |(I)2,2,1,3 | ≤ C|B|1,∞,ê |ˆ u|1,ê ˆ v1 eˆ.

For the second term in (4.37), a Taylor expansion of B11 about any fixed point (ˆ x0 , yˆ0 ) ∈ eˆ gives 1 1 ∂ ˆ ∂ u−u ˆ )1 (ˆ (4.39) (I)2,2,1,2 = φ(ˆ x)B11 (ˆ x0 , yˆ0 ) (Πˆ x, yˆ) vˆ1 (ˆ x, yˆ) dˆ x dˆ y + R, ∂ x ˆ ∂ x ˆ 0 0 where (4.40)

u|1,ê ˆ v1 eˆ, |R| ≤ C|B|1,∞,ê |ˆ

using (4.7) for the last inequality. To bound the first term on the right-hand side in (4.39), we note that 1 (φ2 ) (ˆ x) = 6φ(ˆ x) + , 2

(φ2 ) (0) = (φ2 ) (1) = 0.


1741

Therefore, using (4.6), we have 1 1 ∂ ˆ ∂ u−u ˆ )1 (ˆ φ(ˆ x)B11 (ˆ x0 , yˆ0 ) (Πˆ x, yˆ) vˆ1 (ˆ x, yˆ) dˆ x dˆ y ∂ x ˆ ∂ x ˆ 0 0 1 1 2 1 ∂ ∂ ˆ ∂ = u−u ˆ )1 (ˆ (φ2 )(ˆ x)B11 (ˆ x0 , yˆ0 ) (Πˆ x, yˆ) vˆ1 (ˆ x, yˆ) dˆ x dˆ y 2 6 ∂ x ˆ ∂ x ˆ ∂ x ˆ (4.41) 0 0 1 1 1 ∂ 2 ∂2 ˆ ∂ =− x)B11 (ˆ x0 , yˆ0 ) 2 (Πˆ u−u ˆ )1 (ˆ x, yˆ) vˆ1 (ˆ x, yˆ) dˆ x dˆ y (φ )(ˆ 6 0 0 ∂x ˆ ∂x ˆ ∂x ˆ ≤ C|B|∞,ê |ˆ u|2,ê ˆ v1 eˆ. A combination of (4.39)–(4.41) gives |(I)2,2,1,2 | ≤ C(|B|1,∞,ê |ˆ u|1,ê + |B|∞,ê |ˆ u|2,ê )ˆ v1 eˆ.

(4.42)

To complete the estimate of (I)2,2,1 , it remains to bound (I)2,2,1,4 . Using that ∂ ˆ u)2 = 0 and (4.34), we have ∂x ˆ (Πˆ (4.43)

1 1 ∂ ∂ ∂ v u ˆ2 vˆ2 dˆ ˆ2 div x dˆ y− φ(ˆ x)B12 u ˆ dˆ x dˆ y ∂ x ˆ ∂ y ˆ ∂ x ˆ 0 0 0 0

1 1 ∂ ∂ ∂ ˆ2 vˆ2 dˆ B12 u ˆ2 vˆ2 dˆ = − φ(ˆ x) B12 u x− φ(ˆ x) x dˆ y ∂x ˆ ∂ yˆ ∂x ˆ 0 0 ˆ3 ˆ1 1 1 ∂ v − φ(ˆ x)B12 u ˆ dˆ x dˆ y. ˆ2 div ∂x ˆ 0 0

(I)2,2,1,4 =

1

1

φ(ˆ x)B12

The last two terms above are bounded by (4.44)

v C[(|B|1,∞,ê |ˆ u|1,ê + |B|∞,ê |ˆ u|2,ê )ˆ v2 eˆ + |B|∞,ê |ˆ u|1,ê div êˆ].

Combining (4.23)–(4.44), we obtain ˆ u)1 vˆ1 = T1 + T2 + T3 , E (BΠˆ (4.45) where (4.46)

|T1 | ≤ C[(|B|2,∞,ê ˆ u1,ê + |B|1,∞,ê |ˆ u|1,ê + |B|∞,ê |ˆ u|2,ê )ˆ veˆ v + (|B|1,∞,ê |ˆ u|eˆ + |B|∞,ê |ˆ u|1,ê )div êˆ],

(4.47)

T2 =

and

ˆ1

T3 =

ˆ3

−

φ(ˆ x) ˆ3

∂ (Bˆ u)1 (ˆ x, yˆ) vˆ2 (ˆ x, yˆ) dˆ x, ∂x ˆ

−

ˆ1

φ(ˆ x) B12 (ˆ x, yˆ)

∂ x, yˆ)ˆ v2 (ˆ x, yˆ) dˆ x. u ˆ2 (ˆ ∂x ˆ

Using Lemma 4.4 below, T1 can be bounded as follows: (4.48) veˆ |T1 | ≤ C h2 K−1 2,∞,e u1,e + hK−1 1,∞,e h|u|1,e + K−1 ∞,e h2 |u|2,e ˆ + hK−1 1,∞,e ue + K−1 ∞,e hu1,e hdiv ve ≤ C h2 (K−1 2,∞,e u2,e ve + K−1 1,∞,e u1,e div ve ),

1742


v using the fact that u|j,ê ≤ Chj uj,e and div êˆ ≤ Chdiv ve (see [13, Lemma 5.5]). |ˆ The term e T2 is treated in Lemma 4.5 below. Finally, term T3 in (4.45) can be rewritten as ∂ ˆ k )ˆ ˆ k dˆ u·n − φ(ˆ s) B12 (ˆ s, yˆ) (ˆ v·n s. T3 = ∂ˆ s ˆ3 ˆ1 A similar term appears in the proof of Theorem 5.1 in [13]. Following the argument there, it can be shown that (4.49) T3 ≤ C h2 u2,e ve . e

e

A combination of estimates (4.45), (4.48), (4.51), and (4.49) implies that ˆ u)1 vˆ1 | ≤ Ch2 (u2 v + u1 div v). |E (BΠˆ e

ˆ u)2 vˆ2 ) is analogous. This completes the proof of the The argument for E((BΠˆ lemma. We next give the proofs of the two auxiliary lemmas used in the above argument. Lemma 4.4. If K−1 ∈ W 2,∞ (Ω), then for all e ∈ Th there exists a positive constant C independent of h such that |B|s,∞,ê ≤ C hs K−1 s,∞,e ,

s = 0, 1, 2.

Proof. First, for a quasi-uniform mesh, we have c1 h ≤ DF∞,ê ≤ c2 h,

c3 h2 ≤ J∞,ê ≤ c4 h2

with some positive constants c1 –c4 . This implies that ˆ −1 ∞,ê . B∞,ê ≤ C K

(4.50)

Second, for an h2 -uniform mesh, we have additional estimates. Let α = (α1 , α2 ), αi ≥ 0, be a double index, and let |α| = α1 + α2 . In the case |α| = 1, the definition of the bilinear mapping (2.4)–(2.6) and (2.2) imply that ∂ˆα DF∞,ê ≤ C h2

and

1 ˆα DF ≤ C. ∂ J ∞,ˆ e

In the case |α| = 2, we have the estimates ∂ˆα DF∞,ê = 0

and

1 ˆα DF ≤ C h. ∂ J ∞,ˆ e

As a result, we get ˆ −1 ∞,ê + ∂ˆα K ˆ −1 ∞,ê ) ∂ˆα B∞,ê ≤ C (hK for |α| = 1 and ˆ −1 ∞,ê + h ∂ˆα−1 K ˆ −1 ∞,ê + ∂ˆα K ˆ −1 ∞,ê ) ∂ˆα B∞,ê ≤ C (h2 K


1743

ˆ −1 = K−1 ◦ F, using the chain rule and ∂ˆα F∞,ê ≤ C h|α| for for |α| = 2. Since K |α| ≤ 2, we obtain ˆ −1 ∞,ê ≤ C h|α| K−1 |α|,∞,e , ∂ˆα K

|α| = 0, 1, 2,

which implies ∂ˆα B∞,ê ≤ C h|α| K−1 |α|,∞,e ,

|α| = 0, 1, 2,

completing the proof. Lemma 4.5. If K−1 ∈ W 2,∞ (Ω), then

(4.51)

T2 = 0,

e

where T2 is defined in (4.47). Proof. Summing over all elements in (4.47), we have ∂ ˆ k dˆ u) · τˆ k )ˆ (4.52) T2 = φ(ˆ s) ((Bˆ v·n s. ∂ˆ s ˆk e e k=1,3

Using (2.7), we have that for any edge , (Bˆ u) · τˆ =

1 ˆ −1 DFˆ DFT K u · ||DF−1 τ = ||(K−1 u) · τ . J

Therefore, using (2.9), the sum in (4.52) becomes ∂ −1 2 (K u) · τ k v · nk ds. T2 = (4.53) |k | φ(s) ∂s k e e k=1,3

Since v ∈ Vh , v · n = 0 on exterior edges and v · n is continuous across interior edges. ∂ The assumed regularity for K and u implies that K−1 u and (K−1 u) are continuous ∂s across interior edges. Note that each interior edge appears twice in the sum in (4.53), which now can be rewritten as a sum of interior edge integrals ∂ T2 = ||2 φ(s) ((K−1 u) · τ )[v · n] ds = 0, ∂s e

where [v · n] denotes the jump in the normal component of v. 5. Superconvergence to the average edge fluxes and at the edge midpoints. We now discuss how the superconvergence result from section 4 can be applied to obtain superconvergence for the computed velocity to the average edge fluxes and at the midpoints of the edges. Define, for any v ∈ (H 1 (Ω))2 , (5.1)

∀e ∈ Th ,

|||v|||2e

=

4 k=1

(5.2)

|||v|||2 =

e∈Th

2 v · nk ds

k

|||v|||2e .

,

1744


Using the well-known property of the Piola transformation [6], ˆ dˆ v · n ds = v (5.3) ˆ·n s ∀ v ∈ (H 1 (Ω))2 , ˆ

and transforming to the reference element and back, it is easy to see that ||| · ||| is a norm on Vh and there exist constants c1 and c2 independent of h such that c1 v ≤ |||v||| ≤ c2 v

∀ v ∈ Vh .

It is clear from (4.2) and (5.3) that |||Πv − v||| = 0 for any v ∈ (H 1 (Ω))2 . Therefore, (5.4)

|||u − uh ||| ≤ |||Πu − uh ||| ≤ c2 Πu − uh ≤ Ch2 u2 ,

using Theorem superconvergence of the computed velocity 4.2. This implies edgewise 1 2 uh · n to || u · n ds in a discrete L -sense. Remark 5.1. The superconvergence result (5.4) implies similar superconvergence for |||u − uh |||M with |||v|||2M =

4

|k |2 (v · nk )2 (mk ),

e∈Th k=1

where mk is the midpoint of k . Our choice of reporting the results in ||| · ||| is motivated by the fact that average fluxes are easier to measure than pointwise values and therefore are of greater practical interest. 6. Numerical experiments. In this section, we present the details of the numerical implementation. Instead of solving saddle point problem (2.10), we reduce it to an equivalent system with a symmetric positive definite matrix using the standard hybridization technique. Let Veh be the restriction of Vh to quadrilateral e and Λh be the space of constant functions over edge . Define ˜h = Veh and Λh = Λh . V e

Note that the normal component of vh ∈ Vh is continuous across interior mesh edges and vh · n = 0 on exterior edges. Therefore, h h h h h h h ˜ ˜ ∈V : (μ , v ˜ · ne )∂e = 0 ∀μ ∈ Λ , V = v e

where ne is the outward normal vector for quadrilateral e. It has been shown by many authors (see, e.g., [6]) that the original formulation ˜ h ×W h ×Λh (2.10) is equivalent to the mixed-hybrid formulation: find (˜ uh , ph , λh ) ∈ V such that (K−1 u ˜h , v ˜h )h,e − (ph , div v ˜h )e + (λh , v ˜h · ne )∂e = 0 (6.1)

e

˜ h, ∀˜ vh ∈ V

(div u ˜ h , wh )e = (f, wh )e

∀wh ∈ W h ,

(μh , u ˜ h · ne )∂e = 0

∀μh ∈ Λh .


System (6.1) can be written in ⎛ M BT ⎜ 0 (6.2) ⎝ B C 0

1745

the matrix form as ⎞⎛ ⎞ ⎛ ⎞ u 0 CT ⎟⎜ ⎟ ⎜ ⎟ 0 ⎠⎝ p ⎠ = ⎝ f ⎠, λ 0 0

where D=

M B

BT 0

is a block-diagonal matrix (after a permutation of columns and rows) with as many blocks as mesh elements. Each block is a 5 × 5 matrix. Therefore, vectors u and p can be explicitly eliminated from (6.2) resulting in a system (6.3)

S λ = b,

where S is a sparse symmetric positive definite matrix. For logically rectangular meshes, S has at most seven nonzero elements in each row and column. Its nonzero entries represent connections between edge-based unknowns belonging to the same cell. Problem (6.3) was solved with the preconditioned conjugate gradient (PCG) method. In the numerical experiments, we used one V-cycle of the algebraic multigrid method [24] as a preconditioner. The stopping criterion for the PCG method was the relative decrease in the norm of the residual by a factor of 10−12 . We solved the boundary problem (1.1) with a known analytic solution p(x, y) = x3 y 2 + x cos(xy) sin(x) and tensor coefficient

K(x, y) =

(x + 1)2 + y 2 −xy

−xy . (x + 1)2

It is pertinent to note here that the superconvergence result established in the previous section for the homogeneous Neumann boundary condition can be extended to the case of general Neumann boundary value problem. In example 1, the computational domain Ω is the unit square. The computational grid is constructed from a uniform rectangular grid via the mapping x(ξ, η) = ξ + 0.06 sin(2πη) sin(2πξ),

y(ξ, η) = η + 0.06 sin(2πη) sin(2πξ),

where 0 < η, ξ < 1, and subsequent random distortion of mesh node positions (see Figure 6.1). The maximum value of the distortion is proportional to the square of the local mesh size; i.e., the resulting grid satisfies assumptions (2.2) and (2.3). We test both Neumann and Dirichlet boundary conditions. The results for the Neumann problem are shown in Table 6.1. The convergence rates were computed using the linear regression for the data in the rows for 1/h = 32, 64, 128, 256. In addition to norm (5.2), we show the convergence rate in the discrete L∞ -norm: 1 h h |||u − u |||∞ = max u · nk ds − u · nk , k |k | k

1746


Fig. 6.1. Examples of meshes used in numerical experiments. Table 6.1 Convergence rates for example 1: Neumann boundary conditions. 1/h 8 16 32 64 128 256 Rate

|||u − uh |||∞ 8.32e-2 2.84e-2 8.84e-3 2.42e-3 6.32e-4 1.61e-4 1.93

|||u − uh ||| 5.47e-2 1.69e-2 4.49e-3 1.14e-3 2.87e-4 7.17e-5 1.99

|||p − ph |||∞ 4.75e-3 1.57e-3 4.40e-4 1.16e-4 2.96e-5 7.49e-6 1.96

|||p − ph ||| 1.45e-3 3.99e-4 1.03e-4 2.59e-5 6.48e-6 1.62e-6 2.00

where the maximum is taken over all mesh edges. The convergence rates for the pressure variable are shown in the following discrete norms: |p(ci ) − ph (ci )|2 |ei | |||p − ph |||2 = ei ∈Th

and |||p − ph |||∞ = max |p(ci ) − ph (ci )|, ei ∈Th

where ci is the geometric center of element ei . The use of the geometric center instead of the mass center is due to the following property of the MFD method. The method is exact for linear solutions when the pressure variable, p(ci ), is evaluated at the geometric center ci [15]. The second-order convergence rate is observed for both the pressure and velocity variables in the discrete L2 - and L∞ -norms. In the case of Dirichlet boundary conditions, a loss of one half order in the convergence rate for the velocity in the L2 -norm is expected (see, e.g., [12, 3]). The convergence rates are shown in Table 6.2. Note that the velocity convergence rate in the L2 -norm is larger than the theoretical bound of O(h1.5 ). However, the convergence rate in the L∞ -norm is only O(h). In example 2, the computational domain Ω consists of three quadrilaterals (see Figure 6.1). A sequence of grids is obtained by uniform refinement of these quadrilaterals. The left bottom corner of the domain is located at the point (1, 0). The results


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Table 6.2 Convergence rates for example 1: Dirichlet boundary conditions. |||u − uh |||∞ 1.50e-1 7.20e-2 4.24e-2 2.39e-2 1.27e-2 6.55e-3 0.90

1/h 8 16 32 64 128 256 Rate

|||u − uh ||| 8.58e-2 2.59e-2 6.97e-3 1.81e-3 4.65e-4 1.19e-4 1.96

|||p − ph |||∞ 5.08e-3 1.64e-3 4.71e-4 1.26e-4 3.26e-5 8.26e-6 1.95

|||p − ph ||| 2.08e-3 5.53e-3 1.42e-4 3.57e-5 8.95e-6 2.24e-6 2.00

Table 6.3 Convergence rates for example 2: Neumann boundary conditions. |||u − uh |||∞ 1.59e-1 5.23e-2 1.72e-2 5.65e-3 1.85e-3 6.06e-4 1.61

1/h 8 16 32 64 128 256 Rate

|||u − uh ||| 1.08e-1 2.79e-2 7.07e-3 1.78e-3 4.45e-4 1.11e-4 2.00

|||p − ph |||∞ 8.84e-3 2.74e-3 8.33e-4 2.26e-4 5.84e-5 1.48e-5 1.94

|||p − ph ||| 5.05e-3 1.21e-3 2.95e-4 7.30e-5 1.82e-5 4.53e-6 2.01

of our numerical experiments are shown in Table 6.3. We realize that the grid is only locally h2 -uniform. However, the second-order convergence rate for the velocity variable in the L2 norm is attained. 7. Conclusion. We have proved the superconvergence estimate for the velocity variable on h2 -uniform quadrilateral grids when the exact integration of velocities is replaced by a novel 4-point quadrature rule. The theoretical results for the full diffusion tensor have been confirmed with numerical experiments. Appendix. Representation of the trapezoidal rule error. Lemma A.1. Let f (x, y) be a function defined on a rectangular domain [a, b] × [c, d]. The trapezoidal rule error

b

d

E(f ) ≡

f (x, y) dx dy − (f )T a

c

can be represented as

b

E(f ) = (d − c) a

(x − a)(x − b) ∂ 2 f (x, c) dx 2 ∂x2 d

+ (b − a)

b

c

d

(y − c)(y − d) ∂ 2 f (a, y) dy 2 ∂y 2

(x − b)(y − d) −

+ a

c

(b − a)(d − c) 4

∂2 f (x, y) dx dy. ∂x∂y

Proof. Define a function g (x, s) ≡ (x − k

s)k+

≡

(x − s)k , x ≥ s, 0, x < s,

1748


where k ≥ 0. The Peano kernel theorem (see [23, Theorem 5.2-3, p. 142]) states that the error of the trapezoidal rule is given by b E(f ) = A2,0 (s)f (2,0) (s, c) ds a

(A.1)

d

A0,2 (t)f (0,2) (a, t) dt

+ c

b

a

where f

(i,j)

(x, y) =

∂ i+j ∂xi ∂y j f (x, y)

A2,0 (s) = E(g 1 (x, s)),

d

A1,1 (s, t)f (1,1) (s, t) ds dt,

+ c

for i, j ≥ 0 and

A0,2 (t) = E(g 1 (y, t)),

A1,1 (s, t) = E(g 0 (x, s)g 0 (y, t)).

Straightforward calculations give

(b − a)(d − c) g(xj , s) 4 a c j=1 b b−a (g(a, s) − g(b, s)) (x − s) dx − = (d − c) 2 s b

(A.2)

4

d

A2,0 (s) =

= (d − c)

g 1 (s, x) dx dy −

(s − a)(s − b) . 2

Similarly, we get (A.3) A0,2 (t) = (b − a)

(t − c)(t − d) 2

and A1,1 (s, t) = (s − b)(t − d) −

(b − a)(d − c) . 4

A substitution of (A.2) and (A.3) into (A.1) completes the proof. REFERENCES [1] T. Arbogast, L. C. Cowsar, M. F. Wheeler, and I. Yotov, Mixed finite element methods on nonmatching multiblock grids, SIAM J. Numer. Anal., 37 (2000), pp. 1295–1315. [2] T. Arbogast, C. N. Dawson, P. T. Keenan, M. F. Wheeler, and I. Yotov, Enhanced cell-centered finite differences for elliptic equations on general geometry, SIAM J. Sci. Comput., 19 (1998), pp. 404–425. [3] T. Arbogast, M. F. Wheeler, and I. Yotov, Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences, SIAM J. Numer. Anal., 34 (1997), pp. 828–852. [4] M. Berndt, K. Lipnikov, J. D. Moulton, and M. Shashkov, Convergence of mimetic finite difference discretizations of the diffusion equation, J. Numer. Math., 9 (2001), pp. 253–284. [5] D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, Cambridge, UK, 1997. [6] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Ser. Comput. Math. 15, Springer-Verlag, New York, 1991. [7] Z. Cai, J. E. Jones, S. F. McCormick, and T. F. Russell, Control-volume mixed finite element methods, Comput. Geosci., 1 (1997), pp. 289–315. [8] J. Campbell and M. Shashkov, A tensor artificial viscosity using a mimetic finite difference algorithm, J. Comput. Phys., 172 (2001), pp. 739–765. [9] S.-H. Chou, D. Y. Kwak, and K. Y. Kim, A general framework for constructing and analyzing mixed finite volume methods on quadrilateral grids: The overlapping covolume case, SIAM J. Numer. Anal., 39 (2001), pp. 1170–1196.


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